Environmental Engineering Reference
In-Depth Information
The van Genuchten (1980) equation defines volumetric
water content between saturation and residual water content
conditions (i.e., normalized volumetric water content):
Van Genuchten (1980) proposed a coefficient-of-air-
permeability equation based on an effective degree-of-
saturation equation that embraces the Mualem (1976)
unsaturated permeability model. The end result is a closed-
form expression for relative air permeability. The relative
coefficient of air permeability is assumed to be a function of
degree of air saturation. The van Genuchten-Mualem (1980)
model for the coefficient of air permeability can be expressed
as follows:
1
n
=
(9.82)
(a
v
ψ)
n
v
)
m
v
(
1
+
where:
a
v
,n
v
,m
v
=
soil fitting parameters for the SWCC,
S
e
)
1
/
2
(
1
S
1
/q
e
)
2
q
k
a
=
k
d
(
1
−
−
(9.84)
n
=
normalized volumetric water content defined
as
n
=
θ
−
θ
r
,
where:
θ
s
−
θ
r
θ
=
any selected volumetric water content,
k
a
=
air coefficient of permeability,
θ
s
=
saturation volumetric water content, and
k
d
=
air coefficient of permeability of the dry soil,
θ
r
=
residual volumetric water content.
q
=
fitting parameter related to the pore-size distribution;
van Genuchten (1980) suggested limiting
q
values to
the range between 0 and 1.0.
The van Genuchten (1980) model describes the SWCC
over the range between saturation and residual suction con-
ditions. The equation uses three fitting parameters; namely,
a
v
,n
v
and
m
v
. Other SWCC equations could also be used in
conjunction with Eqs. 9.79 and 9.80 to write equations for
the SWCCs. However, only the Fredlund and Xing (1994)
equation and the van Genuchten (1980) equation are used
to illustrate the estimation procedure.
The van Genuchten (1980) air permeability equation can
be written by substituting Eq. 9.82 into Eq. 9.84. Ba-Te
et al., (2005) extended the van Genuchten (1980) model
(i.e., Eq. 9.84) for use with the Fredlund and Xing (1994)
equation which accepts the entire degree-of-saturation range
(i.e., zero to 100%):
S)
1
/
2
(
1
S
1
/q
)
2
q
k
a
=
k
d
(
1
−
−
(9.85)
9.9.2 Relationships between Air Coefficient
of Permeability and Degree of Air Saturation
A number of empirical relationships have been proposed
for the coefficient of air permeability and the amount of
air in a soil. Published air permeability relationships fall
into two main categories, namely, those that assume power
functions for the air content and those based on the SWCC
(Stylianou and DeVantier, 1995). Power relations include
equations proposed by Irmay (1954), Wyllie (1962), Falta
et al. (1989), and Delage et al. (1998). Relations based on
the SWCC include equations proposed by Brooks and Corey
(1966), Parker et al., (1987), and Cary et al., (1989).
Brooks and Corey (1964) proposed the following empiri-
cal relationship relating the coefficient of air permeability
k
a
to the effective degree of water saturation
S
e
. The equation
applies for soil suctions greater than the air-entry value of
the soil:
9.9.3 Relationship between Air Coefficient
of Permeability and Soil Suction
The Fredlund and Xing (1994) air permeability model and
the van Genuchten (1980) model are used to illustrate
the application of the SWCC to the computation of the
coefficient-of-air permeability function. The Fredlund and
Xing (1994) SWCC equation can be used in conjunction
with Eq. 9.85 to compute the air permeability function. The
van Genuchten (1980) SWCC equation can be modified
and the degree of saturation can be written as follows:
θ
r
θ
r
1
+
e
1
e
S
=
e
−
+
(9.86)
e
[1
+
(a
v
ψ)
n
v
]
m
v
1
+
where:
θ
r
=
residual volumetric water content and
S
e
)
2
(
1
S
(
2
+
λ)/λ
e
k
a
=
k
d
(
1
−
−
)
(9.83)
e
=
void ratio.
The Fredlund and Xing (1994) SWCC equation can be
written as
where:
θ
s
1
S
e
=
effective degree of saturation with respect to the
water phase, defined as
S
e
=
1
+
e
ln
(
1
+
ψ/ψ
r
)
1
S
=
−
(
1
.
0
−
S)/(
1
.
0
−
S
r
)
,
(ψ/a
f
)
n
f
]
m
f
(9.87)
10
6
/ψ
r
)
e
ln
(
1
+
{
ln[
e
+
}
where degree of saturation is in decimal form,
S
r
=
residual degree of saturation,
where:
k
d
=
coefficient of air permeability when the soil is dry
(i.e., zero water content), and
e
=
void ratio,
λ
=
fitting coefficient.
θ
s
=
saturated volumetric water content,
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